Convective diffusion high Reynolds numbers

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

Axial and radial dispersion coefficients are equal at low Reynolds numbers because the dispersion is due to the molecular diffusion and the axial and radial structures of the bed are similar (Gunn, 1968). However, at high Reynolds numbers, the convective dispersion dominates and the values are different because the axial dispersion is primarily caused by differences in the fluid velocity in the flow channels, whereas the radial dispersion is primarily caused by deviations in the flow path caused by the particles. 

Flat plate. Let us investigate convective diffusion to the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers. The velocity field of the fluid near a flat plate is presented in Subsection 1.7-2. We assume that mass transfer is accompanied by a surface reaction. 

Zhang J (1997) Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number. Numer Methods PDEs 13 77-92... 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

The consequence of a large Schmidt number, common in liquids, is that convection dominates over diffusion at moderate and even relatively low Reynolds numbers (assuming consistent order of magnitude in the terms). In gases these effects are of the same order. On the other hand, heat transfer in low-viscosity liquids by convection and conduction are the same order since the Prandtl number is approximately 1. In highly viscous fluids where the Prandtl number is large, heat transfer by convection predominates over conduction, provided the Reynolds number is not small. The opposite is true for liquid metals, where the Prandtl number is very small, so conduction heat transfer is dominant. 

Since for liquids Sc 1, it follows from (5.116) that Peu 1 for values Re > 1, that is the convective flux dominates over the diffusion in liquids at finite (and sometimes at small) Reynolds numbers. In high-viscous liquids, Pr 1 and from (5.115) it follows that Pej 1 at not very small numbers Re. Hence, in this case the heat transfer is basically due to convection. 

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